INVERSE HYPERBOLIC PROBLEM:

UNIQUENESS AND STABILITY

Masahiro Yamamoto

Department of Mathematical Sciences The University of Tokyo 3-8-1 Komaba, Meguro, Tokyo 153 Japan e-mail : myama@ms.u-tokyo.ac.jp

With this lecture, we will discuss inverse hyperbolic problems of determining functions in the equations. We presently take the following initial/boundary value problem for a single hyperbolic equation of the second order: (1.1) ρ(x)∂_{t}^{2}u(x, t) = Δu(x, t)+p(x)∂_{t}u(x, t)−q(x)u(x, t)+f(x)R(x, t), x ∈ Ω, 0 < t < T.

(1.2)  u(x, 0), ∂_{t}u(x, 0), x ∈ Ω : given.

(1.3)  the Dirichlet case : u_{|∂Ω×(0,T)} is given.

(1.4)  the Neumann case: \frac{∂u}{∂ν}_{|∂Ω×(0,T)} is given.

Here and henceforth, let Ω ⊂ ℝ^{n} (n ≥ 1) be a bounded domain whose boundary ∂Ω be of class C^{2}. We set ∂_{t} = \frac{∂}{∂t}, ∂_{t}^{2}u = \frac{∂^{2}}{∂t^{2}} , ∂_{i} = \frac{∂}{∂x_{i}}, Δ = ∑_{i=1}^{n} ∂_{i}^{2}, ∇ = (∂_{1}, ..., ∂_{n}), ν = ν(x) the outward normal vector to ∂Ω at x and \frac{∂}{∂ν} = ∇ · ν, R is given.

In this report, we survey

Inverse hyperbolic problem with a single measurement. Let D ⊂ \overline{Ω} be suitably given and let T > 0. Determine some (or one) of the functions ρ, p, q and f from observations u_{|D×(0,T)} or \frac{∂u}{∂ν}_{|D×(0,T)}.

In this report, we will exclusively discuss formulation with a single measurement. For the mathematical point of view, we can point the following two topics for the inverse problem: the uniqueness and the stability.

The main purpose of this report is to show recent results on the uniqueness and the stability which have been obtained by the author and his colleagues.

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